While On The Base Leg Of The Airport Traffic Pattern The Right Main Landing Gear Did Not Fully Extend Analysis: Both pilots reported that after performing airwork they returned to the departure airport where multiple uneventful full stop landings were performed. They then departed and while on the base leg of the airport traffic pattern the right main landing gear did not fully extend. At that point the designated pilot examiner (DPE) took the controls from the pilot applicant, and during the course of 90 minutes, made multiple attempts to fully lower the right main landing gear that were ultimately unsuccessful.